If `int (dx)/(sin^6 x + cos^6 x) = tan^(-1) f(x)`, then

Options

A. `f(x) = tan x - cot x`
B. `f(pi/4) = 0`
C. `f(x) = tan x + cot x`
D. `f(pi/4) = 2`

Answer

To solve the integral \( I = \int \frac{dx}{\sin^6 x + \cos^6 x} \) and express it in the form \( \tan^{-1} f(x) \), we will follow these steps:

### Step 1: Simplify the Denominator
We start with the expression in the denominator:
\[ \sin^6 x + \cos^6 x \]
Using the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \), we can rewrite it as:
\[ \sin^6 x + \cos^6 x = (\sin^2 x + \cos^2 x)(\sin^4 x - \sin^2 x \cos^2 x + \cos^4 x) \]
Since \( \sin^2 x + \cos^2 x = 1 \), we have:
\[ \sin^6 x + \cos^6 x = \sin^4 x - \sin^2 x \cos^2 x + \cos^4 x \]

### Step 2: Further Simplification
Next, we can express \( \sin^4 x + \cos^4 x \) using the identity:
\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - 2\sin^2 x \cos^2 x \]
Thus, we have:
\[ \sin^6 x + \cos^6 x = 1 - 3\sin^2 x \cos^2 x \]

### Step 3: Substitute \( t = \tan x \)
Now, we will perform the substitution \( t = \tan x \), which gives us \( dx = \frac{dt}{1+t^2} \) and \( \sin^2 x = \frac{t^2}{1+t^2} \), \( \cos^2 x = \frac{1}{1+t^2} \). Therefore:
\[ \sin^2 x \cos^2 x = \frac{t^2}{(1+t^2)^2} \]
Substituting these into the integral:
\[ \sin^6 x + \cos^6 x = 1 - 3\frac{t^2}{(1+t^2)^2} \]
Thus, the integral becomes:
\[ I = \int \frac{1+t^2}{(1 - 3\frac{t^2}{(1+t^2)^2})(1+t^2)} dt = \int \frac{(1+t^2)^2}{1+t^2 - 3t^2} dt = \int \frac{(1+t^2)^2}{1 - 2t^2} dt \]

### Step 4: Simplify the Integral
Now we simplify the integral:
\[ I = \int \frac{(1+t^2)^2}{1 - 2t^2} dt \]
We can perform polynomial long division or split the integral into simpler parts.

### Step 5: Integrate
We can integrate term by term:
\[ I = \int \left( \frac{1}{1 - 2t^2} + \frac{2t^2}{1 - 2t^2} \right) dt \]
The first part can be integrated using the formula for the integral of a rational function, and the second part can be integrated using substitution.

### Step 6: Back Substitute
After integrating, we will back substitute \( t = \tan x \) to express the result in terms of \( x \).

### Step 7: Compare with \( \tan^{-1} f(x) \)
Finally, we will compare our result with \( \tan^{-1} f(x) \) to determine \( f(x) \).